Well-Posedness for Low Regularity Solutions to the g-SQG Equation with Regular Level Sets

We show that the generalized SQG equation on the plane is locally well-posed in spaces of low regularity solutions (essentially Hölder continuous with Hölder exponents depending on the equation parameter α∈(0, 12)) that have H² level sets (i.e., with L² curvatures). Moreover, for α≤ 16 and initial data satisfying some additional hypotheses we show that the corresponding solutions can stop existing only when their level sets lose H²-regularity, and hence not just due to level set collisions or "pile ups".